How many ways can 4 math books and 5 English books be put on a shelf if all the math books and the English books have to put together?

Question: "In how many ways can 2 different history books, 5 different math books, and 4 different novels be arranged on a shelf if the books of each type must be together?"

In this question, sequence of the books is not important, therefore:

• For the 2 history books: 2 ways to arrange them (AB and BA), or $2!$
• For the 5 math books: $5*4*3*2*1 = 5!$ ways to arrange them, or 120
• For the 4 novels: $4*3*2*1 = $4!$ ways to arrange them, or 24

Think like this:

• For the history books (assuming we only look at the history books): 2 options for the first slot, and 1 for the last
• For the math books (again, only look at the math books): 5 options for the first slot, $5-1=4$ for the second slot, $5-2=3$ for the third and so on
• The same for the novels

We also have three types of books, so, the order of first-to-appear is, by the same logic, 3!

Therefore, in the the end you have $2!*5!*4!*3!=34560$ ways to arrange those books

Answer:

5,760 ways

Step-by-step explanation:

Let x and y be the English and Math books.

Since same books must be grouped together, let's see first how many ways can we arrange the English books.

x = 5 × 4 × 3 × 2 × 1 = 120

Now let's see how many ways can we arrange the Math books.

y = 4 × 3 × 2 × 1 = 24

Now let's see how many ways can we arrange the two groups of books.

2 × 1 = 2

Let's multiply all the possible ways to get the final answer.

120 × 24 × 2 = 5,760

Therefore, we can arrange the 5 English books and 4 Math books in 5,760 ways on a shelf if books of the same subject are to be together.

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